A Lagrangian Neighbourhood Theorem for shifted symplectic derived schemes

Pantev, Toen, Vaquié and Vezzosi arXiv:1111.3209 defined k-shifted symplectic derived schemes and stacks X for k∈Z, and Lagrangians f:L→X in them. They have important applications to Calabi-Yau geometry and quantization. Bussi, Brav and Joyce arXiv:1305.6302 proved a 'Darboux Theorem' giving explicit Zariski or étale local models for k-shifted symplectic derived schemes X for k<0 presenting them as twisted shifted cotangent bundles. We prove a 'Lagrangian Neighbourhood Theorem' giving explicit Zariski or etale local models for Lagrangians f:L→X in k-shifted symplectic derived schemes X for k<0, relative to the Bussi-Brav-Joyce 'Darboux form' local models for X. That is, locally such Lagrangians can be presented as twisted shifted conormal bundles. We also give a partial result when k=0. We expect our results will have future applications to k-shifted Poisson geometry (see arXiv:1506.03699), to defining 'Fukaya categories' of complex or algebraic symplectic manifolds, and to categorifying Donaldson-Thomas theory of Calabi-Yau 3-folds and 'Cohomological Hall algebras'.